Finite Group Databasebeta

← #373790

Subgroups — #373790

Order 512.

1586 conjugacy classes of subgroups.

The subgroup lattice has 625 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2354C211yesmaximal, characteristic
2C2×#1948C211yesmaximal, characteristic, complemented
3C2×#1804C211yesmaximal, characteristic
4C2×#1762C211yesmaximal, characteristic
5#14150C211yesmaximal, characteristic, complemented
6#11121C212yesmaximal
7#11066C212yesmaximal
8#10908C212yesmaximal, complemented
9#10884C212yesmaximal, complemented
10#10843C211yesmaximal, characteristic, complemented
11#10835C211yesmaximal, characteristic, complemented
12C22×C4×D4C2211yescharacteristic
13C22×#117—21no—
14C22×#106C2212yes—
15C22×C4⋊C8C2211yescharacteristic
16C22×#102C2211yescharacteristic
17C22×#102C2211yescharacteristic
18C22×#102C2212yes—
19C22×#97C2211yescharacteristic
20#2354—24no—
21#2354—24no—
22C2×#390C2212yes—
23C2×#389C2212yes—
24C2×#388—22no—
25C2×#384C2211yescharacteristic
26C2×#382C2211yescharacteristic
27C2×#380—21no—
28C2×C4×C4⋊C4C2212yes—
29C2×C4×C4⋊C4C2211yescharacteristic
30C2×C4×C4⋊C4C2211yescharacteristic
31#1948—24no—
32#1804C418yescomplemented
33#1762—24no—
34C2×#340C2212yes—
35C2×#340C2212yes—
36C2×#340C2211yescharacteristic
37C2×#340C2211yescharacteristic
38C2×#340C2212yes—
39C2×#340C2212yes—
40C2×#336C2212yes—
41C2×#336C2212yes—
42#1444—21no—
43#1442—21no—
44#1436—22no—
45C4⋊#95C2212yes—
46C23×D4C2311yescharacteristic
47C24×C4—21no—
48C23×C8C2312yes—
49C23×C8C2311yescharacteristic
50C23×C8C2311yescharacteristic
51C2×C4×D4—22no—
52C2×C4×D4—22no—
53C2×C4×D4—22no—
54C2×C4×D4C2×C414yes—
55C2×C4×D4D414yes—
56C2×C4×D4Q814yes—
57C2×#117—28no—
58C2×#117—48no—
59C22×C4⋊C4—21no—
60C22×C4⋊C4—21no—
61C22×C4⋊C4—22no—
62C22×C4⋊C4D411yescharacteristic
63C22×C4⋊C4D411yescharacteristic
64C22×C4⋊C4C2312yes—
65C22×C4⋊C4C2312yes—
66C22×C4⋊C4C2311yescharacteristic
67C22×C4⋊C4C2311yescharacteristic
68C22×C4⋊C4C2311yescharacteristic
69C22×C22⋊C4—21no—
70C22×C22⋊C4—22no—
71C22×C22⋊C4—22no—
72C22×C42—22no—
73C22×C42—21no—
74C22×C42C2312yes—
75C22×C42C2311yescharacteristic
76C2×#106—24no—
77C2×#106—24no—
78C2×#106—42no—
79C2×C4⋊C8—22no—
80C2×C4⋊C8—22no—
81C2×C4⋊C8C2×C414yes—
82C2×#102—22no—
83C2×#102—22no—
84C2×#102—22no—
85C2×#102—22no—
86C2×#102—22no—
87C2×#102—22no—
88C2×#102—42no—
89C2×#102—42no—
90C2×#102C2×C418yes—
91C2×#97—22no—
92C2×#97—22no—
93C2×#97—22no—
94#390—24no—
95#390—24no—
96#389—24no—
97#389—42no—
98#388—216no—
99#384—22no—
100#384D414yes—
101#382—22no—
102#382C2×C414yes—
103#380—28no—
104C4×C4⋊C4—24no—
105C4×C4⋊C4—24no—
106C4×C4⋊C4—22no—
107C4×C4⋊C4—22no—
108C4×C4⋊C4—22no—
109C4×C4⋊C4—22no—
110C2×#95—21no—
111C2×#95—21no—
112C2×#95—22no—
113C2×#95—22no—
114C2×#95—22no—
115C2×#95—21no—
116C2×#95—22no—
117C2×#95D412yes—
118#340—24no—
119#340—24no—
120#340—22no—
121#340—22no—
122#340—42no—
123#340C2×C418yes—
124#336—24no—
125#336—42no—
126C25—21no—
127C22×D4—21no—
128C22×D4—21no—
129C22×D4—21no—
130C22×D4—21no—
131C22×D4—21no—
132C22×D4—21no—
133C22×D4—21no—
134C22×D4—21no—
135C22×D4—21no—
136C22×D4—21no—
137C22×D4—21no—
138C22×D4C22×C412yes—
139C22×D4C2×D412yes—
140C22×D4C2×Q812yes—
141C23×C4—21no—
142C23×C4—24no—
143C23×C4—24no—
144C23×C4—24no—
145C23×C4—22no—
146C23×C4—21no—
147C23×C4—22no—
148C23×C4—21no—
149C23×C4—44no—
150C23×C4—44no—
151C23×C4C2×D412yes—
152C23×C4C2×D412yes—
153C23×C4C2×D411yescharacteristic
154C23×C4C2×D411yescharacteristic
155C23×C4C2411yescharacteristic
156C22×C8—21no—
157C22×C8—21no—
158C22×C8—22no—
159C22×C8—22no—
160C22×C8—21no—
161C22×C8—21no—
162C22×C8—21no—
163C22×C8—21no—
164C22×C8—21no—
165C22×C8—21no—
166C22×C8—22no—
167C22×C8—22no—
168C22×C8—21no—
169C22×C8—22no—
170C22×C8—21no—
171C22×C8—22no—
172C22×C8—21no—
173C22×C8—21no—
174C22×C8C22×C414yes—
175C22×C8#4112yes—
176C22×C8#4112yes—
177C4×D4—28no—
178C4×D4—28no—
179C4×D4—28no—
180C4×D4C4⋊C4116yescomplemented
181#117—432no—
182C2×C4⋊C4—22no—
183C2×C4⋊C4—24no—
184C2×C4⋊C4—21no—
185C2×C4⋊C4—21no—
186C2×C4⋊C4—21no—
187C2×C4⋊C4—21no—
188C2×C4⋊C4—21no—
189C2×C4⋊C4—21no—
190C2×C4⋊C4—21no—
191C2×C4⋊C4—22no—
192C2×C4⋊C4—22no—
193C2×C4⋊C4—22no—
194C2×C4⋊C4—21no—
195C2×C4⋊C4—21no—
196C2×C4⋊C4—22no—
197C2×C4⋊C4—22no—
198C2×C4⋊C4—22no—
199C2×C4⋊C4—22no—
200C2×C4⋊C4—21no—
201C2×C4⋊C4—22no—
202C2×C4⋊C4—21no—
203C2×C4⋊C4—24no—
204C2×C4⋊C4—22no—
205C2×C4⋊C4—22no—
206C2×C4⋊C4—22no—
207C2×C4⋊C4—21no—
208C2×C4⋊C4—21no—
209C2×C4⋊C4—24no—
210C2×C4⋊C4—42no—
211C2×C4⋊C4—44no—
212C2×C4⋊C4—44no—
213C2×C4⋊C4—44no—
214C2×C4⋊C4—82no—
215C2×C4⋊C4—82no—
216C2×C4⋊C4C22×C414yes—
217C2×C4⋊C4C22×C412yes—
218C2×C4⋊C4C2×D414yes—
219C2×C4⋊C4C2×D412yes—
220C2×C4⋊C4C2×Q812yes—
221C2×C4⋊C4#4112yes—
222C2×C4⋊C4#4112yes—
223C2×C22⋊C4—24no—
224C2×C22⋊C4—24no—
225C2×C22⋊C4—24no—
226C2×C22⋊C4—28no—
227C2×C22⋊C4—28no—
228C2×C22⋊C4—44no—
229C2×C22⋊C4—44no—
230C2×C22⋊C4—82no—
231C2×C22⋊C4—82no—
232C2×C42—28no—
233C2×C42—22no—
234C2×C42—22no—
235C2×C42—22no—
236C2×C42—22no—
237C2×C42—22no—
238C2×C42—22no—
239C2×C42—44no—
240C2×C42—44no—
241C2×C42—42no—
242C2×C42—82no—
243C2×C42C22×C412yes—
244C2×C42C2×D412yes—
245C2×C42C2×D412yes—
246C2×C42C2×Q812yes—
247C2×C42#4112yes—
248C2×C42#4112yes—
249#106—48no—
250C4⋊C8—28no—
251#102—28no—
252#102—28no—
253#102—48no—
254#97—44no—
255#95—28no—
256#95—24no—
257#95—24no—
258#95—24no—
259#95—44no—
260#95—44no—
261#95—42no—
262#95—82no—
263C24—22no—
264C24—22no—
265C24—22no—
266C24—41no—
267C24—41no—
268C24—41no—
269C24—41no—
270C24—42no—
271C24—42no—
272C24—42no—
273C24—42no—
274C24C22×D411yescharacteristic
275C2×D4—22no—
276C2×D4—22no—
277C2×D4—22no—
278C2×D4—22no—
279C2×D4—22no—
280C2×D4—22no—
281C2×D4—22no—
282C2×D4—22no—
283C2×D4—22no—
284C2×D4—22no—
285C2×D4—22no—
286C2×D4—22no—
287C2×D4—22no—
288C2×D4—22no—
289C2×D4—22no—
290C2×D4—41no—
291C2×D4—41no—
292C2×D4—41no—
293C2×D4—41no—
294C2×D4—41no—
295C2×D4—41no—
296C2×D4—41no—
297C2×D4—41no—
298C2×D4—41no—
299C2×D4—41no—
300C2×D4—41no—
301C2×D4—41no—
302C2×D4C2×C4⋊C414yes—
303C22×C4—22no—
304C22×C4—22no—
305C22×C4—22no—
306C22×C4—22no—
307C22×C4—22no—
308C22×C4—22no—
309C22×C4—22no—
310C22×C4—22no—
311C22×C4—22no—
312C22×C4—22no—
313C22×C4—22no—
314C22×C4—22no—
315C22×C4—22no—
316C22×C4—216no—
317C22×C4—24no—
318C22×C4—22no—
319C22×C4—24no—
320C22×C4—22no—
321C22×C4—22no—
322C22×C4—22no—
323C22×C4—22no—
324C22×C4—22no—
325C22×C4—22no—
326C22×C4—22no—
327C22×C4—22no—
328C22×C4—48no—
329C22×C4—416no—
330C22×C4—416no—
331C22×C4—48no—
332C22×C4—42no—
333C22×C4—42no—
334C22×C4—41no—
335C22×C4—41no—
336C22×C4—42no—
337C22×C4—41no—
338C22×C4—42no—
339C22×C4—42no—
340C22×C4—42no—
341C22×C4—41no—
342C22×C4—42no—
343C22×C4—42no—
344C22×C4—42no—
345C22×C4—42no—
346C22×C4—42no—
347C22×C4—42no—
348C22×C4—42no—
349C22×C4—42no—
350C22×C4—42no—
351C22×C4—42no—
352C22×C4—42no—
353C22×C4—42no—
354C22×C4C4×D412yes—
355C22×C4C4×D412yes—
356C22×C4#12012yes—
357C22×C4#12012yes—
358C22×C4C22⋊Q812yes—
359C22×C4C22⋊Q812yes—
360C22×C4AGL(1,ℤ/8)14yes—
361C22×C4C23×C411yescharacteristic
362C22×C4C22×D411yescharacteristic
363C22×C4C22×D411yescharacteristic
364C22×C4C22×Q811yescharacteristic
365C22×C4C2×#4111yescharacteristic
366C22×C4C2×#4111yescharacteristic
367C22×C4C2×#4111yescharacteristic
368C22×C4C2×#4111yescharacteristic
369C22×C4C2×#4111yescharacteristic
370C22×C4C2×#4111yescharacteristic
371C22×C4H(5,𝔽2)11yescharacteristic
372C22×C4H(5,𝔽2)11yescharacteristic
373C22×C42−1+411yescharacteristic
374C22×C42−1+411yescharacteristic
375C2×C8—24no—
376C2×C8—24no—
377C2×C8—24no—
378C2×C8—22no—
379C2×C8—22no—
380C2×C8—22no—
381C2×C8—22no—
382C2×C8—22no—
383C2×C8—22no—
384C2×C8—41no—
385C2×C8—41no—
386C2×C8—41no—
387C2×C8—41no—
388C2×C8—42no—
389C2×C8—42no—
390C2×C8—42no—
391C2×C8—42no—
392C2×C8—41no—
393C2×C8—41no—
394C2×C8—41no—
395C2×C8—41no—
396C4⋊C4—28no—
397C4⋊C4—22no—
398C4⋊C4—22no—
399C4⋊C4—22no—
400C4⋊C4—24no—
401C4⋊C4—24no—
402C4⋊C4—22no—
403C4⋊C4—22no—
404C4⋊C4—22no—
405C4⋊C4—24no—
406C4⋊C4—24no—
407C4⋊C4—22no—
408C4⋊C4—44no—
409C4⋊C4—48no—
410C4⋊C4—42no—
411C4⋊C4—41no—
412C4⋊C4—41no—
413C4⋊C4—42no—
414C4⋊C4—41no—
415C4⋊C4—41no—
416C4⋊C4—42no—
417C4⋊C4—42no—
418C4⋊C4—84no—
419C4⋊C4—88no—
420C4⋊C4C2×C4⋊C414yes—
421C22⋊C4—216no—
422C22⋊C4—416no—
423C22⋊C4—88no—
424C42—416no—
425C42—42no—
426C42—42no—
427C42—42no—
428C42—42no—
429C42—84no—
430C42C2×C4⋊C414yes—
431C42C4×D414yes—
432C42C22⋊Q814yes—
433C42#12314yes—
434C23—24no—
435C23—44no—
436C23—44no—
437C23—42no—
438C23—42no—
439C23—42no—
440C23—42no—
441C23—44no—
442C23—42no—
443C23—44no—
444C23—42no—
445C23—42no—
446C23—42no—
447C23—42no—
448C23—42no—
449C23—82no—
450C23—82no—
451C23—82no—
452C23—82no—
453C23—82no—
454C23—82no—
455C23—82no—
456C23—82no—
457C23C2×C4×D411yescharacteristic
458C23C2×#12011yescharacteristic
459C23C2×C22⋊Q811yescharacteristic
460C23#54611yescharacteristic
461C23#54711yescharacteristic
462C23#54811yescharacteristic
463C23#55011yescharacteristic
464C23C2×AGL(1,ℤ/8)12yes—
465C23#57512yes—
466C23#57712yes—
467C23#57712yes—
468D4—24no—
469D4—24no—
470D4—42no—
471D4—42no—
472D4—42no—
473D4—42no—
474D4—42no—
475D4—42no—
476C2×C4—24no—
477C2×C4—24no—
478C2×C4—24no—
479C2×C4—24no—
480C2×C4—22no—
481C2×C4—22no—
482C2×C4—22no—
483C2×C4—22no—
484C2×C4—24no—
485C2×C4—24no—
486C2×C4—24no—
487C2×C4—24no—
488C2×C4—432no—
489C2×C4—44no—
490C2×C4—42no—
491C2×C4—42no—
492C2×C4—44no—
493C2×C4—42no—
494C2×C4—42no—
495C2×C4—42no—
496C2×C4—42no—
497C2×C4—42no—
498C2×C4—42no—
499C2×C4—44no—
500C2×C4—44no—
501C2×C4—42no—
502C2×C4—42no—
503C2×C4—42no—
504C2×C4—44no—
505C2×C4—44no—
506C2×C4—44no—
507C2×C4—44no—
508C2×C4—42no—
509C2×C4—42no—
510C2×C4—44no—
511C2×C4—44no—
512C2×C4—42no—
513C2×C4—42no—
514C2×C4—82no—
515C2×C4—82no—
516C2×C4—82no—
517C2×C4—82no—
518C2×C4—82no—
519C2×C4—82no—
520C2×C4—82no—
521C2×C4—82no—
522C2×C4—81no—
523C2×C4—81no—
524C2×C4—81no—
525C2×C4—81no—
526C2×C4—82no—
527C2×C4—82no—
528C2×C4#38514yes—
529C2×C4#38514yes—
530C2×C4C22×C4⋊C411yescharacteristic
531C2×C4C2×C4×D411yescharacteristic
532C2×C4C4×#4111yescharacteristic
533C2×C4C4×#4111yescharacteristic
534C2×C4#51811yescharacteristic
535C2×C4#51911yescharacteristic
536C2×C4C2×C22⋊Q811yescharacteristic
537C2×C4C2×#12311yescharacteristic
538C2×C4#52511yescharacteristic
539C2×C4#52511yescharacteristic
540C2×C4#53311yescharacteristic
541C2×C4#53311yescharacteristic
542C2×C4#53711yescharacteristic
543C2×C4#53711yescharacteristic
544C2×C4#54411yescharacteristic
545C2×C4#54411yescharacteristic
546C2×C4#54611yescharacteristic
547C2×C4#54711yescharacteristic
548C2×C4#54811yescharacteristic
549C2×C4#55011yescharacteristic
550C2×C4#55211yescharacteristic
551C2×C4#55211yescharacteristic
552C2×C4#55311yescharacteristic
553C2×C4#55311yescharacteristic
554C2×C4#55511yescharacteristic
555C2×C4#55511yescharacteristic
556C2×C4#55611yescharacteristic
557C2×C4#55611yescharacteristic
558C8—44no—
559C8—42no—
560C8—42no—
561C22—44no—
562C22—44no—
563C22—44no—
564C22—84no—
565C22—82no—
566C22—84no—
567C22—82no—
568C22—82no—
569C22—82no—
570C22—84no—
571C22—84no—
572C22C2×#38511yescharacteristic
573C22#236211yescharacteristic
574C22#236311yescharacteristic
575C22#240711yescharacteristic
576C22#240711yescharacteristic
577C22#243911yescharacteristic
578C22#244111yescharacteristic
579C22C4×AGL(1,ℤ/8)12yes—
580C22#295212yes—
581C22#300311yescharacteristic
582C22#300511yescharacteristic
583C22#300911yescharacteristic
584C22#301011yescharacteristic
585C22#307712yes—
586C22#308412yes—
587C22#330612yes—
588C22#332112yes—
589C22#334512yes—
590C22#334812yes—
591C22#336812yes—
592C22#337612yes—
593C22#339712yes—
594C22#340112yes—
595C4—24no—
596C4—24no—
597C4—44no—
598C4—44no—
599C4—82no—
600C4—84no—
601C4—82no—
602C4—82no—
603C4—84no—
604C4#229611yescharacteristic
605C4#231611yescharacteristic
606C4#236211yescharacteristic
607C4#236311yescharacteristic
608C4#243911yescharacteristic
609C4#244111yescharacteristic
610C4#248511yescharacteristic
611C4#248511yescharacteristic
612C2—84no—
613C2#1412911yescharacteristic
614C2#2087311yescharacteristic
615C2#2088211yescharacteristic
616C2#2307212yes—
617C2#2310912yes—
618C2#2412811yescharacteristic
619C2#2413311yescharacteristic
620C2#2419711yescharacteristic
621C2#2420011yescharacteristic
622C2#2603212yes—
623C2#2609912yes—
subgroupG/H
center, socleC24C22×D4
commutatorC22×C4C23×C4
Frattini, cosocleC23×C4C24
prime radicalC23×D4C23
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C24 ◁ C2×C4×C4⋊C4 ◁ G
LECSG ▷ C23×C4 ▷ C2 ▷ C1

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

NH ≅ G/Ntype
2. C2×#1948C2semidirect
5. #14150C2semidirect
8. #10908C2semidirect
9. #10884C2semidirect
10. #10843C2semidirect
11. #10835C2semidirect
32. #1804C4semidirect
180. C4×D4C4⋊C4semidirect

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroupindex
C2×#23542
C2×#19482
C2×#18042
C2×#17622
#141502
#111212
#110662
#109082
#108842
#108432
#108352